Solve for $x$ and $y$ using elimination. $\begin{align*}-2x+5y &= 6 \\ -2x+9y &= 8\end{align*}$
Answer: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}2x-5y &= -6\\ -2x+9y &= 8\end{align*}$ Add the top and bottom equations. $4y = 2$ Divide both sides by $4$ and reduce as necessary. $y = \dfrac{1}{2}$ Substitute $\dfrac{1}{2}$ for $y$ in the top equation. $-2x+5( \dfrac{1}{2}) = 6$ $-2x+\dfrac{5}{2} = 6$ $-2x = \dfrac{7}{2}$ $x = -\dfrac{7}{4}$ The solution is $\enspace x = -\dfrac{7}{4}, \enspace y = \dfrac{1}{2}$.